Index 467 AA, 216 right triangle, 236 SAS, 218 SSS, 218 similarity, transitive property of, 214 simple polygonal region, 199 simple statement, 384 sine, 238 sines, law of, 241 single elliptic geometry, 35, 41, 136 singleton, 424 slope of a line, 33 small angle lemma, 325 SMSG (School Mathematics Study Group), 21, 379 SMSG postulates, 379–380 south pole, 35 specification in Greek proofs, 7 to define a set, 424 spherical geometry, 34, 41, 135 spherical trigonometry, 371 square, 175, 194 area of, 206 constructing, 196, 300, 305 diagonal of, 234 square geometry, 31 square of a number, 434 square region, 199 square root, 435, 438 square root of 2 is irrational, 5 squaring a circle, 308, 317 a figure, 304 a polygon, 304 a rectangle, 304 SSS congruence theorem, 107, 111 SSS construction theorem, 235 SSS existence theorem, 234 SSS similarity theorem, 218 standard measure of an angle, 87 starting point of a ray, 71 statement atomic, 384 compound, 385 existential, 397 mathematical, 383 simple, 384 universal, 394 straight angle, 84 straight line, 2 straightedge, 3, 295, 296 strong passing point, 174 subfield, 309 subpolygons cut off by a chord, 162 subset, 424 substitution property of equality, 393 such that in existence statements, 397 in logical conjunctions, 385 sufficient condition, 389 sum formula for cosine, 241 for sine, 241 summit angles, 341 summit of a Saccheri quadrilateral, 341 superposition, method of, 15 superset, 424 supplementary angles, 91 supremum, 280 surjective function, 56, 429 symmetric property of equality, 393 symmetry, 372 of a Klein geometry, 373 of asymptotic rays, 347 of betweenness of points, 60 of betweenness of rays, 89 of equidistant lines, 188 symmetry group, 373 tangent circles, 253 tangent circles theorem, 254 tangent line, 249 existence and uniqueness, 252, 258 exterior to circle, 252 properties of, 252 tangent line theorem, 251 tangent segment to a circle, 265 tangential polygon, 265, 268 is convex, 266 tangential quadrilateral, 269, 276 tangential triangle theorem, 268 taxicab distance, 138 taxicab geometry, 138 Thales of Miletus, 257 Thales’ postulate, 335 Thales’ theorem, 257 converse, 258 theorem, 6, 23, 405 in an axiomatic system, 23 in Euclid’s Elements, 6 there exists (existential quantifier), 397 three-point line, 31

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